jacoby, stangeland and wajeeh, 20001 valuation of bonds and stock ufirst principles: u value of...

Jacoby, Stangeland and Wajeeh, 2000

1

Valuation of Bonds and Stock

First Principles: Value of financial securities = PV of expected

future cash flows

To value bonds and stocks we need to: Estimate future cash flows:

size (how much) and timing (when) Discount future cash flows at an appropriate

rate

Chapter 5


2

Bond Features

What is a bond - debt issued by a corporation or a governmental body. A bond represents a loan made by investors to the issuer. In return for his/her money, the investor receives a legal claim on

future cash flows of the borrower.

The issuer promises to: make regular coupon payments every period until the bond

matures, and pay the face (par) value of the bond when it matures.

Default an issuer who fails to pay is subject to legal action on behalf of the

lenders (bondholders).

3

Pure-Discount (Zero-Coupon) Bonds Information needed for valuing pure discount bonds:

Time to maturity (T):

T = Maturity date - today’s date Face value (F) Discount rate (r)

0 1 2 … T

|-------------------|-------------------|------ … ------|

F Value of a pure discount bond:

PV = F / (1 + r)T

4

Examples - Pure Discount BondsQ1. Consider a zero-coupon bond, with a face value of $1,000,

maturing in 5 years. Suppose that the appropriate discount rate is 8%. What is the current value of the bond?

A1. This is a simple TVM problem:

Use the above PV equation to solve:

PV = F / (1 + r)T = 1,000 / (1.08)5 = $

Q2. Suppose 6 months have past. What is the bond value now?

A1. Again, use the above PV equation to solve:

PV = F / (1 + r)T = 1,000 / (1.08)4.5 = $

Note: As we get closer to maturity(T), the z.c. bond value increases (PV), since we have to wait less time to receive $1,000

0 1 2 3 4 5Year:(r = 8%)

1,000PV0

5

Level-Coupon Bonds Information needed to value level-coupon bonds:

Coupon payment dates and Time to maturity (T) Coupon (C) per payment period and Face value (F) Discount rate

0 1 2 … T

|----------------|------------------|------- … ------|

Coupon Coupon Coupon + F

Value of a Level-coupon bond:

PV = C/(1+r) + C/(1+r)2 + .. + C/(1+r)T + F/(1+r)T

= C (1/r){1 - [1 / (1 + r)T]} + F/(1 + r)T

= PV of coupon payments + PV of face value

6

Example - Coupon BondsQ1. Consider a coupon bond paying a 4% coupon rate annually, with a

face value of $1,000, maturing in 10 years. Suppose that the appropriate discount rate is 6%. What is the current value of the bond?

A1. The time line:

Define:

c = annual coupon rate (%)

C = dollar periodic coupon payment = cFIn the above example:

c = % C = cF = = $

F = $ T = years r = %


PV= C (1/r){1 - [1 / (1 + r)T]} + F/(1 + r)T = 40(1/0.06){1 - [1 / (1.06)10]} + 1,000/(1.06)10 = $

0 1 2 9 10

(Years)(r = 6%)…

7

First, clear previous data, and check that your calculator is set to 1 P/YR:

The display should show: 1 P_Yr

Input data (based on above bond example)

PV of a Bond in your HP 10B Calculator

PMT

I/YR

N

PV

Key in coupon payment

Key in discount rate

Key in number of periods to maturity

Compute PV of the bond

Display should show: -852.79825897

40

6

10

FVKey in face value (paid at maturity)

1,000

YellowC

C ALL

8

Example - Discount, Premium and Par BondsQ2. For the above coupon bond: when discount rate is 6% and

coupon rate is 4% (c < r), the value of the bond is $852.80, less than its face value (PV < F). In this case we say that the bond is priced at discount. Recalculate the PV of the above bond with discount rates of 2% and 4%.

A2. r = 2%

We have: r = 2% < 4% = c.


PV= C (1/r){1 - [1 / (1 + r)T]} + F/(1 + r)T = 40(1/0.02){1 - [1 / (1.02)10]} + 1,000/(1.02)10 =

$1,179.65

We see that when c > r, the bond is priced at premium (PV > F).

r = 4%

We have: r = c = 4%.


PV= 40(1/0.04){1 - [1 / (1.04)10]} + 1,000/(1.04)10 = $1,000

We say that when c = r, the bond is priced at par (PV = F).


9

Some Tips on Bond Pricing

Bond prices and market interest rates move in opposite directions.

When coupon rate = market rate (r) => price = par value.

(par bond)

When coupon rate > market rate (r) => price > par value (premium bond)

When coupon rate < market rate (r) => price < par value (discount bond)


10

PV

r (%)

1,000 = F

4 = c2 6

($)

Premium Bond(r < c , and PV>F)

Discount Bond(r > c , and PV<F)

Par Bond(r = c , and PV=F)

Discount, Premium, and Par Bonds

11

The bond’s indenture provides: F, c, and T

The bond price (B) is set by the market

Given F, c, T, and B, what return (y) does the market demand for holding the bond?

To find y, solve the following equation:

There is no analytical solution (use calculator)

TT

TT

yF

yy

yF

yC

yC

yC

CB

B

)1()1(11

)1()1()1()1(

1

or ,21

Yield-To-Maturity


12

Example - Bond’s YTMQ. Consider a coupon bond paying a 7% coupon rate annually, with a

face value of $1,000, maturing in 20 years. The current market price of the bond is $1,072.93. What is the yield to maturity (YTM) of the bond?

A. We have:

c = % C = cF = =$

F = $ T = B = $

Use your HP 10B Financial Calculator:

PMT

PV

N

I/YR

1) Key in coupon

payment

3) Key in the

bond price (PV)

4) Key in number of

periods to maturity

5) Compute YTM Display should show: 6.346178%

70

1,072.93

20

FV2) Key in face value 1,000

+/-

YellowC

C ALL

13

Canadian BondsCanadian bonds usually pay coupons every six months (semiannually)

Q. Consider a GofC bond paying semiannual coupons at an annual rate of

6%, with a face value of $1,000, maturing in 8 years. The bond’s YTM is

7% per year compounded semiannually. What is the value of the bond?

A. The time line:

Define:

c = (stated) annual coupon rate (%) = 6%

C = dollar periodic coupon payment = (c/2)F= = $

We also have:

F = 1,000 N = s.a. periods y1/2 = 7% per year comp. s.a.

Use the following PV equation to solve:

0 1 2 15

16

(6-month Periods)(y1/2 = 7% per year comp. s.a.)

…

53.939$130

)1()1(

11

1

16207.016

207.0

207.0

2/12/12/1

)1(

000,1

)1(11

222

NyNyy

FcPV

14

First, clear previous data, and check that your calculator is set to 1 P/YR:

The display should show: 1 P_Yr

Input data (based on above bond example)

PV of a S.A. Coupon Bond in your HP 10B Calculator

PMT

I/YR

N

PV

Key in the s.a. coupon payment

Key in the effective s.a. discount rate

Key in number of 6-month periods to maturity

Compute PV of the bond

Display should show: -939.52941596

30

3.5

16

FVKey in face value (paid at maturity)

1,000

YellowC

C ALL

15

Finding the YTM of Canadian Bonds

Q. Consider a GofC bond paying semiannual coupons at an annual rate of

12%, with a face value of $1,000, maturing in 25 years. The bond’s market value is $1,057.98. What is the yield to maturity (YTM) of the bond per year compounded semiannually?

A. We have: c = 12% C = (c/2)F= = $

F = 1,000 N = s.a. periods B= 1,057.98


Since (y1/2/2) = 5.649995%, the YTM of the bond per year compounded

semiannually is given by: y1/2 = 25.649995% = 11.299990%

PMT

PV

N

I/YR

1) Key in s.a. coupon

payment

3) Key in the

bond price (PV)

4) Key in number of s.a.

periods to maturity

5) Compute the

effective YTM

PER 6 MONTHS

Display should show: 5.649995%

60

1,057.98

50


+/-YellowC

C ALL

16

Finding the Maturity of Canadian BondsQ. Consider a GofC bond paying semiannual coupons at an annual rate of

7%, with a face value of $1,000. The bond’s market value is $1,026.82, with a YTM of 6.6% per year compounded semiannually. What is the time to maturity of this bond in years?

A. We have: c = 7% C = (c/2)F= = $

F = 1,000 (y1/2 /2) = = % B=$


Thus, T = 0.5(# of 6-month periods to maturity)

= 0.517.99806470 = 8.99903235 years to maturity

PMT

PV

I/YR

N

1) Key in s.a. coupon

payment

3) Key in the

bond price (PV)

4) Key in the effective

YTM per 6 months

5) Compute the number

of 6 MONTHS

PERIODS to maturity

Display should show: 17.99806470

35

1,026.82

3.3


+/-Yellow

CC ALL

17

Finding the Coupon Rate of Canadian Bonds

Q. Consider a 30-year GofC bond paying semiannual coupons, with a face value of $1,000. The bond’s market value is $912.83, with a YTM of 8.4% per year compounded semiannually. What is the the bond’s annual coupon rate?

R. We have: F = 1,000 (y1/2/2) = = %

B= 912.83 N = s.a. periods


Since: C = (c/2)F, we get:

c = (C/F)2 = (38.00001999/1,000)2 = 7.600004% per annum

N

PV

I/YR

PMT

1) Key in number of s.a.

periods to maturity

3) Key in the

bond price (PV)

4) Key in the effective

YTM per 6 months

5) Compute the SEMI

ANNUAL DOLLAR

coupon payment

Display should show: 38.00001999

60

912.83

4.2


+/-Yellow

CC ALL


18

From the financial Post, November 28, 1997Years to Bond

Maturity maturity yields (%)

1-Dec-97 0.00 0.001-Nov-98 0.92 4.571-Mar-99 1.25 4.701-Dec-99 2.00 4.981-Mar-00 2.25 5.021-Sep-00 2.75 5.091-Mar-01 3.25 5.161-Dec-01 4.00 5.241-Apr-02 4.33 5.251-Jun-03 5.50 5.371-Dec-03 6.00 5.411-Jun-04 6.50 5.421-Dec-04 7.01 5.471-Dec-05 8.01 5.531-Dec-06 9.01 5.561-Jun-10 12.51 5.731-Mar-11 13.25 5.7615-Mar-14 16.30 5.841-Jun-21 23.52 5.93

The Yield Curve

4

4.5

5

5.5

6

6.5

0 5 10 15 20 25 30

Years to maturity

Bon

d yi

elds

(%

)

19

Four theories:

I. Expectations theory

e.g. if investors expect next years yield to be 12%, then

the forward rate will also be 12%: 1f2 = 12%

Example: two alternative investments B1: a zero coupon bond with: T = 1, YTM: 0r1 = 8%

B2: a zero coupon bond with: T = 2, YTM: 0r2 = 9%,

The Term Structure of Interest Rates

1

1 over year expected ratespot

tt

tt

rE

tf


20

The following investments of $1 must be equivalent:

(i) investing in B2 for 2 years. At t = 2, receive: 1.092

(ii) investing in B1 for 1 year. At t = 1, investing in a new

1-year bond at a rate 1f2. At t = 2, receive: 1.08(1+1f2)

The forward rate (1f2) must take a value such that:

The Term structure of Interest Rates

)1(08.1)09.1( 212 f

21

This implies the following forward rate for year-2:

The general case:

Note: This formula can be used only for zero-coupon bonds

Example: 3 alternative zero-coupon bonds, with the following spot rates:

0r1 = 8%

0r2 = 10%

0r3 = 12%

Calculating 1f2 and 2f3:

%01.1011 08.109.1

)1()1(

2122

10

20

rrf

1)1(

)1(1

0

01

1

tt

tt

r

rtt f

%037.1211 08.110.1

)1(

)1(21

2

1

2

10

20

r

rf

%110.1611 2

3

2

3

10.112.1

)1(

)1(32

20

30

r

rf

22

Example - Using the Term StructureQ1. You observe the above spot rates for GofC zero-coupon bonds for

different maturities: 0r1 = 8%, 0r2 = 10%, and 0r3 = 12%. A zero- coupon bond has a face value of $1,000 and maturity of 2 years. What must be its price today?

A1. Since: (1+0r2)2 = (1+0r1)(1+1f2), we can use either spot rates or

forward rates (same result) to find B:

Q2. Assume that the Pure Expectations Hypothesis (PEH) holds, what do you expect the bond price to be one year from today?

A2. One year from today, the bond will have one year remaining to maturity. Based on the PEH:

expected spot rate for second year = 1f2

Thus, the expected bond price in a year is:

45.826$:

:or ,45.826$:

12037.108.1000,1

)1)(1(000,1

)1.1(

000,1

)1(

000,1

2110

2022

fr

r

BForward

BSpot

56.892$][ 12037.1000,1

)1(000,1

121

fBE

21rE

23

II. Liquidity Premium Theory

If you invest for (t+1) years, you commit to reinvest in every

year after the 1st year, and thereby lose liquidity and ask

for a liquidity premium:

III. Augmented Expectations Theory

Combines the pure expectations theory with the liquidity premium theory:

Example - Suppose: 0r1 = 8% and 0r2 = 9%. By the Expectations Theory:

By the Liquidity Premium Theory, when L2 =1%, we get:

1f2= E[1r2] + L2. = E[1r2] + 1%

Both theories together, give:

11

11 )1(over year expected ratespot

ttt

ttt

LrE

Ltf

2121 rEf

)1(08.109.1 212 f

)01.01(08.109.1 212 rE


24

Shapes of the Term Structure under the Liquidity Premium Theory & the Augmented Expectations Theory

The shape of the term structure depends on the magnitude of the premium: when there exist constant expectations:

when there exist decreasing expectations:

%

t

f ``

f `

t

% f

E[trt+1]

L

E[trt+1]

25

IV. Market Segmentation Theory

Different segments of investors choose to invest in assets with different

investment horizons:

e.g. mutual funds that commit to invest in long term bonds.

Thus, demand and supply in each segment could set different rates:

t

%

Short TermBonds

Medium TermBonds

Long TermBonds


26

Common Stocks

What are stocks - legal representation of of ownership in a corporation (equity) a stock holder is entitled to receive profit distributions of the

corporation (dividends)

Dividends: cash payments made by the corporation to stockholders since stocks have no expiration date, we assume that dividends will

be paid forever

Valuation the value of stocks at any point in time equals the present value of

all future dividends


27

Common Stock Valuation

The value of a stock = PV of all expected future cash flows

Thus, the information needed to value common stocks: Common Stock Dividends (Dt) Discount rate (r)

PV0 = D1/(1 + r)1 + D2/(1 + r)2 + D3/(1 + r)3 + . . . forever. .

We have to estimate future dividends


28

Case 1: Zero Growth

Assume that dividends will remain at the same level forever, i.e. D1 = D2 =…= Dt = D

Since future cash flows are constant, the value of a zero growth stock is the present value of a perpetuity:

Pt = Dt+1 / r


29

Example - Valuation of Common Stocks with Zero Growth

Q. ABC Corp. is expected to pay $0.75 dividend per annum, starting a year from now, in perpetuity. If stocks of similar risk earn 12% annual return, what is the expected price of a share of ABC stock?

A. The stock price is given by the the present value of the perpetual stream of dividends:

P0 = D1 / r

= =

0 1 2 3 4

$ $ $ $


30

Case 2: Constant Growth Assume that dividends will grow at a constant rate, g, forever, i. e.,

D1 = D0 x (1+g)

D2 = D1 x (1+g) = D0 x (1+g)2

Dt = D0 x (1+g)t

Since future cash flows grow at a constant rate forever, the value of

a constant growth stock is the present value of a growing perpetuity:

Pt = Dt+1 / (r - g)


31

Examples - Valuation of Common Stocks with Constant Growth

Q1. XYZ Corp. has a common stock that paid its annual dividend this morning. It is expected to pay a $3.60 dividend one year from now, and following dividends are expected to grow at a rate of 4% per year into the foreseeable future (forever)in perpetuity. If stocks of similar risk earn 16% effective annual return, what is the price of a share of XYZ stock?

A1. The stock price is given by the the present value of the perpetual stream of growing dividends:

P0 = D1 / (r-g)

= =

0 1 2 3 4

$3.60 $3.60 $3.60 $3.60

forever . . .

32

Q2. In the above example, assume that XYZ’s common stock that paid its quarterly dividend two months ago. It is expected to pay a $0.90 dividend

in one month, and following quarterly dividends are expected to grow at

a rate of 1% per quarter into the foreseeable future. Recall that the

effective annual required rate of return on XYZ stock is 16%. What is the

price of a share of XYZ stock now?

A2. Time line of the quarterly dividends:

We first need to calculate EPR1/4 and EPR1/12:

Using: EPRn = (1+EAR)n - 1, we get:

EPR1/4 = 3.780199% and EPR1/12 = 1.244514%

The stock price in one month (after D1 month is paid):

P1 month = D4 months / (EPR1/4 - g)

= (0.901.01)/(0.03780199 - 0.01) = $32.69550129

The stock price today:

P0 = (D1 month + P1 month) / (1+EPR1/12)

= (0.90+ 32.69550129) / 1.01244514 = $33.18

0 1 month 4 months

7 months 10 months

$0.90 $0.90 $0.90 $0.90

forever . . .

33

Q3. Manitoba Network Operators (MNO) is expected to pay a dividend next year of $8.06 per share. Both sales and profits for Pale Hose are expected to grow at a rate of 2% per year indefinitely. Its dividend is expected to grow by the same amount. If an investor is currently willing to pay $62.00 per one MNO share, what is her required return for this investment?

A3. We have: P0=$62.00, D1=$8.06, and g=0.02. We are looking for r.

The stock price is given by:

P0 = D1/(r-g)

=

Rearranging, we get:

r = =

In general:

r = (D1/P0) + g

34

Q4. Vandalay Industries Corp. (VIC) is expected to pay a dividend next year of $4.32 per share. Its current stock price is $36. If therequired return for this stock is 15%, what is the constant dividend growth rate expected for VIC’s stock starting from the second year forever?

A4. We have: P0=$36.00, D1=$4.32, and r=0.15. We are looking for g.


P0 = D1/(r-g)

=


g = =

In general:

g = r - (D1/P0)


35

Q5. MT&T Inc. has a common stock that paid its annual dividend this morning. You expect future annual dividends to grow at a rate of 2% per year into the foreseeable future (forever). The required return for this stock is 20%, and its current price is $25.50. What is the dividend that was paid this morning?

A5. We have: P0=$25.50, r=0.20, and g=0.02. We are looking for D0.


P0 = D1/(r-g)

25.5 = D1/(0.20-0.02)


D1 = 25.5(0.20-0.02) = $4.59

We are looking for D0. Since D1=D0(1+g), D0 is given by:

D0 = D1/(1+g) = = $

36

Case 3: Differential Growth

Assume that dividends will grow at different rates in the foreseeable future and then will grow at a constant rate thereafter.

To value a Differential Growth Stock, we need to: Estimate future dividends in the foreseeable

future. Estimate the future stock price when the stock

becomes a Constant Growth Stock (case 2). Compute the total present value of the

estimated future dividends and future stock price at the appropriate discount rate.

37

Examples - Differential GrowthQ1. Whizzkids Inc. is experiencing a period of rapid growth. Earnings and

dividends are expected to grow at a rate of 8 percent during the next three years, and then at a constant rate of 4% thereafter. Whizzkids’ last dividend,which has just been paid, was $2 per share. If the required rate of return onthe stock is 12 percent, what is the price of the stock today?

A1. It is given that:

r = 12%, D0 = $2, g1 = g2 = g3 = 8%, and g4 = g* = 4% (forever)We calculate:

D1=$2 =$ , D2= =$ ,

D3= =$

With g4=g* =4%, we have:

D4= =$

Since constant growth rate applies to D4, we use Case 2 (constant growth) to compute P3:

P3 = = $


38

Expected future cash flows of this stock:

0 1 2 3 |----------|---------|---------| (r = 12%)

D1 D2 D3 + P3

2.16 2.33 2.52 + 32.75

The current (time 0) value of the stock:

P0 = D1/(1+r) + D2/(1+r)2 + (D3+P3)/(1+r)3

= + +

= $

39

Q2. An investor has just paid $141.75 for the purchase of one share of UMB

Corp. stock. UMB just paid a $9 dividend per share. Annual dividends paid

at the end of the first, second and third years will grow at a rate of 10% per

annum, and then grow at a constant annual rate of g* forever. Given the risk

inherent in UMB Corp., the investor requires an effective annual rate of

10% on his/her investment. What is the value of g*?

A2. We calculate:

D1=$9=$ , D2= =$ ,

D3= =$

With g4=g*, we have:

D4=

UMB’s current stock price is given by:

P0 = D1/(1+r) + D2/(1+r)2 + (D3+P3)/(1+r)3

Where: P3 = D4/(r-g*)


40

With the above data:

141.75 = 9.9/1.1 + 10.89/(11)2 + 11.979/(11)3 + P3/(1.1)3

Thus, the expected stock price in three years is P3 = 152.73225

Since, this price is given by:

P3 = D4/(r-g*) = [D3(1+g*)]/(r-g*)

We have

152.73225 =


g* = %

jacoby, stangeland and wajeeh, 20001 valuation of bonds and stock ufirst principles: u value of...

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